Transmission network



Feb. 16, 1937. E. L. NORTON TRANSMISSION NETWORK Filed June 9, 1954 n w m 6 4 ifs L'a FREQUENCY ATTORNV jatented Feb. 16, 1937 UNITED STATES PATENT OFFICE Bell Telephone Laboratories,

Incorporated,

New York, N. Y., a corporation of New York Application June 9, 1934, Serial No. '729,743

3 Claims.

This invention relates to frequency selective networks and more particularly to the control of the phase characteristics of broad-band transmission systems. It has for its principal object the provision of a linear phase characteristic, not only in the transmission band of a band selective system, but also in the frequency ranges adjacent to the band wherein the attenuation is rapidly increasing. A further object is to provide for the control of the rate, at which the attenuation increases at the edges of the transmission band while at the same time maintaining the linearity of the phase characteristic in these regions.

The control of the transmission characteristics of a reactive four-terminal network through the allocation of the resonance and anti-resonance frequencies of the branch impedance is described in United States Patent 1,828,454, to- H. W. Bode, wherein particular frequency distributions are disclosed which provide linear variation of the phase characteristic throughout substantially the whole transmission band of the wave lter.

The characteristics with which the above mentioned patent deals are those of the network itself without reference to the impedances between which it is connected, namely, the characteristic impedance and the propagation constant. When such networks are connected to operate between xed terminal impedances, which Ain practice are usually fixed resistances, the overall transmission characteristics of the system include not only the propagation constant of the network itself, but also the effects of Wave reflection at the junctions to the terminal impedances. These latter effects may be negligibly small throughout the greater part of the band when the characteristic impedance is a substantially constant resistance, but in the frequency ranges adjacent to the band limits, the wave reflection exercises a strong modifying influence.

In the networks of the invention, the reflection effects for a preassigned Value of the terminal resistances are coordinated with the propagation constant in such a manner that the total phase characteristic of the system represented by the network and its connected terminal impedance is linear throughout the whole of the transmission band and part of the attenuation ranges ad.- jacent thereto. This is accomplished b-y allocating the resonance and anti-resonance frequencies of the branch impedances of the networks in accordance with a novel arrangement as hereinafter described, one characteristic of the arrangement being that the coincidence of the resonance and anti-resonance frequencies of the different network branches, which has heretofore been regarded as necessary for the pro- (Cl. 178-44l insertion loss due to the lattice will be given by where I1 is the current through the receiving-end resistance with the network absent and I2 is the corresponding current with the network in the circuit. Expressing I1 and 12in terms of E and the impedances and substituting in the above equation, the insertion loss may be written as Assuming that the impedances are pure re- Vactances, and writing Equation 2 may be transformed to l-I-tan y1 tan y2 l-l-tan y1 tan y2 If we write 0=A7`B, where A is an attenuation factor and B is the phase angle, then Substituting in the preceding equation and equating real and imaginary terms we will have l-tan y1 tan y2 It is thus seen that the phase angle B is equal to the sum of the angles y1 and y2, which are determined by the reactance arms X1 and X2 according to Equations 1 and 2; and that the attenuation A is determined by the difference between these angles. When y1 and y2 are nearlyvequal the attenuation of the network is small and when their difference approaches 90 degrees the attenuation approaches ini'inity. It is to be noted that in the foregoing discussion noassumption has been made as to the coincidence of the'resonance and anti-resonance frequencies of the reactances X1 and X2 as is usually done in the design of filters of this type. A

The application of the foregoing principles to the design of networks of the invention will now be described. To obtain linearity of the phase characteristic through a given frequency range it is necessary to provide reactive impedances Z1 and Z2 such that the sum of the angles y1 and y2, defined above, increases linearly with frequency through the range. If this linearity is to be accompanied by broad band selective properties it is further necessary that the impedances be such as to make the diiference between y1 and y2 small throughout the transmission band and approximately 90 degrees elsewhere. A graphic method of design may be used which is described below in connection with the design of a low pass filter.

In Fig. 3 the curves represent the variations of the network constants with frequency, the abscissae being proportional to the ratio f/o where f a denotes frequency and fo is arfrequency of the character of a cut-off frequency marking the limit of the transmission band. Curve 3 represents a desired phase characteristic, the ordinates being taken for convenience as equal to the angle B/2. Curve 4 represents the desired attenuation characteristic, the ordinates being in decibels attenuation, corresponding to the scale on the right hand side of the gure. It is to be noted that the desired phase characteristic is substantially linear up to a frequency equal to 1.3] and that the attenuation is less thank 4 decibels at all frequencies below fo but rises sharply to infinity at the frequency 1.3fo.

The design now proceeds by drawing in the curves representing the angles y1 and y2 using the above mentioned principles as a guide. In the first place it is clear that y1 and y2 must be evenly spaced about curve 3 since their sum is to be equal to twice the ordinates of curve 3.

' In the second place, the differencelbetween the ordinates of the two curves is determined by the desired attenuation loss in accordance with the formula 21:20 logm (1o) tenuation is 4 decibels, the difference of the angles y1 and y2 is 51 degrees or .284 1r radians.

Having determined the curves representing y1 and y2 the next step is to determine the reactances corresponding to these characteristics in accordance with Equation 4. From these equations it is seen that the reactance X1 will be zero at zero frequency and at those frequencies for Which the ordinates of curve I are even multiples of 1r/2 and will be infinite at frequencies for which the ordinates are odd multiples of 11/2. In the case of reactance X2 an 4inverse relationship holds. For the case illustrated in Fig. 3, the critical frequencies of reactance X1 occur at zero, 0.33510, 0.68fo, .99fo, and 1.3fo, and those of X2 at Zero, 0.3810, 0.7510, and 1.3fo. 'Ifhe next critical frequency in each case is assumed to beiniinity, that is, the reactances have no further resonances or anti-resonances at finite frequencies and the curves of y1 and yz'iiatten out as the frequency increases. The frequency characteristic of the reactances are shown in Fig. 4, curve 5 corresponding to X1 and curve 6 to reactance X2. The lack of coincidence of the critical frequencies is clearly shown. It will be observed that the increasing rslope of curve I in Fig. 3 as the frequency increases entails a progressive diminution of the intervals between critical frequencies of the reactance X1 and that the diminishing slope of curve 2 entails a progressive increase in the intervals between the critical frequencies of reactance X2. This is illustrated in Fig. ,4 by the increasing lack of coincidence between the resonances, or zeros, of curve 6, and the anti-resonances, or poles, of curve 5. zeros of X2 and the poles of X1 are `nearly coincident, but separate gradually as thevattenuation increases until in the range of high attenuation the zeros of the two reactances substantially coincide. In the example illustrated each reactance has only one critical frequency in the attenuating range, but as many as desired may be added. Since the curves I and 2 must continue to be separated by approximately 1r/2 ra- 1 dians to `maintain high attenuation, the location of the added critical frequencies must be such as to maintain the curves substantially parallel.

The physical character of reactance X1 is illusr In the range of low attenuation the trated by the impedance Z1 of Fig. 2 and that of X2 by the impedance Z2 of the same figure. The former consists of two resonant circuits and a simple inductance all connected in parallel and the latter of a capacity and an inductance and 1 an anti-resonant loop all in series.

It is well known that, theoretically, the lack of lcoincidence of the critical frequencies should 'produce a multi-bandl system. Thus the network of Fig. 2, with the critical frequencies indicated above, should have two attenuation bands below fo in the ranges adjacent the first antiresonance and the first resonance 'of X1 where the reactances X1 and X2 are of like sign. However, the amount of attenuation in those bands v is quite small an-d by virtue of the reiiection effects at the terminal impedances is made unnoticeable. All of the effects are summed up in Equations` 8 and 9 which show that the attenuation is small so long as the difference of y1 and y2 is small.

Having ascertained the critical frequencies of the reactances by the graphic method outlined above the values of the several inductances and capacities can be ascertained by the method described by R. M. Foster in an article entitled A Reactance Theorem, Bell System Technical Journal, Vol. III, No. 2, April 1924. Before the formulae given in this article can be applied, however, it is necessary that one element in each impedance be determined. 'Ihis can be done most conveniently by considering the slopes of the y1 and y2 characteristics at zero frequency, which may be made each equal to the slope of the curve 3, that is, to half the slope of the desired linear phase characteristic.

From Equation 4 the slopes of the y1 and y2 characteristics are found to be At zero frequency the reactance of Z1 is evidently wholly that of the simple shunt inductance L12 and the reactance of Z2 is wholly determined by the series capacity C22. Hence if So be the desired linear slope of the resultant phase characteristics, Equations 13 and 14 become and 'SOZZTI'RDCM (16) which suffice to determine the elements Lia and C22.

It will be observed that the phase characteristic illustrated by curve 3 of Fig. 3 undergoes a sudden shift of vr/2 at the frequency 1.33%), representing a phase shift of 1r, or a current reversal, in the over-all characteristic. A discontinuity of this type will occur Whenever the attenuation passes through an innite value, as at 1.310, but it is not of great consequence since the current that is reversed is innitesimally small. Linearity of the phase characteristic is maintained up to the rst attenuation peak above the cut-off frequency and, if reversals be ignored, can be maintained as far as may be desired by increasing the complexity of impedances Z1 and Z2 so that they have additional critical frequencies above the cut-off.

The invention has been described in connection with its embodiment in a broad band selective network, that is, a network characterized by a steep transition between the transmission and the attenuation ranges. It is obvious from the discussion given that the design principles may be applied in other connections, for example, in the design of a network giving a gradually rising or gradually falling attenuation coupled with phase linearity and useful as an attenuation equalizer.

What is claimed is:

1. A four-terminal broad-band wave filter com-v prising a symmetrical network of reactances, said network having characterizing reactances X1 and X2 which by their frequency variations and magnitudes determine the transmission properties of the network, equal resistive terminal impedances of resistance Rn connected to the input and the output terminals of said lter, said characterizing reactances each having a plurality of critical frequencies corresponding to resonances and anti-resonances, said critical frequencies being so spaced in the frequency scale that the sum of the angles y1 and y2 dened by y1=tan1R` and y2 tan X2 varies substantially linearly with frequency throughout the transmission band of the filter and throughout part of the attenuation range adjacent the band.

2. A wave filter in accordance with claim l in which the critical frequencies and magnitudes of the reactances X1 and X2 are so proportioned that the angles y1 and y2 are zero at zero frequency, are substantially equal in a low frequency range, and diverge in a higher frequency range to a difference equal to 1r/2 radians, their sum being proportional to the frequency throughout both ranges, whereby the filter has a low-pass characteristic and a linear phase shift throughout and beyond the transmission range.

3. A wave filter in accordance with claim 1 of the low-pass type in which the critical frequencies of the reactances X1 and X2 are approximately uniformly spaced in the lower portion of the transmission band whereby the angles y1 and y2 are approximately equal and proportional to frequency and in which at higher frequencies the intervals of the critical frequencies of X1 diminish while the intervals of the corresponding frequencies of X2 increase so that over a preassigned frequency range the difference of the angles y1 and y2 increases gradually to an approximately constant value of 1r/2 radians, the average value of the twoangles retaining the same proportionality to frequency as in the lower range.

EDWARD L. NORTON. 

